metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD32⋊3D5, D10.6D8, D8.5D10, C16.11D10, Q16.2D10, C40.19C23, C80.11C22, Dic5.25D8, D40.3C22, Dic20.4C22, C4.7(D4×D5), (D5×C16)⋊5C2, C5⋊3(C4○D16), C5⋊D16⋊4C2, C5⋊Q32⋊2C2, C16⋊D5⋊6C2, C2.22(D5×D8), D8⋊3D5⋊5C2, (C5×SD32)⋊4C2, (C4×D5).60D4, C20.13(C2×D4), C10.38(C2×D8), C5⋊2C8.26D4, Q8.D10⋊4C2, (C5×D8).5C22, C8.25(C22×D5), C5⋊2C16.6C22, (C8×D5).41C22, (C5×Q16).3C22, SmallGroup(320,543)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD32⋊3D5
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a7, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 422 in 84 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C10, C16, C16, C2×C8, D8, D8, SD16, Q16, Q16, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C16, D16, SD32, SD32, Q32, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4○D16, C5⋊2C16, C80, C8×D5, D40, Dic20, D4.D5, Q8⋊D5, C5×D8, C5×Q16, D4⋊2D5, Q8⋊2D5, D5×C16, C16⋊D5, C5⋊D16, C5⋊Q32, C5×SD32, D8⋊3D5, Q8.D10, SD32⋊3D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, SD32⋊3D5
►On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 105)(2 112)(3 103)(4 110)(5 101)(6 108)(7 99)(8 106)(9 97)(10 104)(11 111)(12 102)(13 109)(14 100)(15 107)(16 98)(17 55)(18 62)(19 53)(20 60)(21 51)(22 58)(23 49)(24 56)(25 63)(26 54)(27 61)(28 52)(29 59)(30 50)(31 57)(32 64)(33 135)(34 142)(35 133)(36 140)(37 131)(38 138)(39 129)(40 136)(41 143)(42 134)(43 141)(44 132)(45 139)(46 130)(47 137)(48 144)(65 92)(66 83)(67 90)(68 81)(69 88)(70 95)(71 86)(72 93)(73 84)(74 91)(75 82)(76 89)(77 96)(78 87)(79 94)(80 85)(113 152)(114 159)(115 150)(116 157)(117 148)(118 155)(119 146)(120 153)(121 160)(122 151)(123 158)(124 149)(125 156)(126 147)(127 154)(128 145)
(1 47 85 18 151)(2 48 86 19 152)(3 33 87 20 153)(4 34 88 21 154)(5 35 89 22 155)(6 36 90 23 156)(7 37 91 24 157)(8 38 92 25 158)(9 39 93 26 159)(10 40 94 27 160)(11 41 95 28 145)(12 42 96 29 146)(13 43 81 30 147)(14 44 82 31 148)(15 45 83 32 149)(16 46 84 17 150)(49 125 108 140 67)(50 126 109 141 68)(51 127 110 142 69)(52 128 111 143 70)(53 113 112 144 71)(54 114 97 129 72)(55 115 98 130 73)(56 116 99 131 74)(57 117 100 132 75)(58 118 101 133 76)(59 119 102 134 77)(60 120 103 135 78)(61 121 104 136 79)(62 122 105 137 80)(63 123 106 138 65)(64 124 107 139 66)
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 129)(28 130)(29 131)(30 132)(31 133)(32 134)(33 63)(34 64)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 95)(74 96)(75 81)(76 82)(77 83)(78 84)(79 85)(80 86)(97 160)(98 145)(99 146)(100 147)(101 148)(102 149)(103 150)(104 151)(105 152)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)
G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,105)(2,112)(3,103)(4,110)(5,101)(6,108)(7,99)(8,106)(9,97)(10,104)(11,111)(12,102)(13,109)(14,100)(15,107)(16,98)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,135)(34,142)(35,133)(36,140)(37,131)(38,138)(39,129)(40,136)(41,143)(42,134)(43,141)(44,132)(45,139)(46,130)(47,137)(48,144)(65,92)(66,83)(67,90)(68,81)(69,88)(70,95)(71,86)(72,93)(73,84)(74,91)(75,82)(76,89)(77,96)(78,87)(79,94)(80,85)(113,152)(114,159)(115,150)(116,157)(117,148)(118,155)(119,146)(120,153)(121,160)(122,151)(123,158)(124,149)(125,156)(126,147)(127,154)(128,145), (1,47,85,18,151)(2,48,86,19,152)(3,33,87,20,153)(4,34,88,21,154)(5,35,89,22,155)(6,36,90,23,156)(7,37,91,24,157)(8,38,92,25,158)(9,39,93,26,159)(10,40,94,27,160)(11,41,95,28,145)(12,42,96,29,146)(13,43,81,30,147)(14,44,82,31,148)(15,45,83,32,149)(16,46,84,17,150)(49,125,108,140,67)(50,126,109,141,68)(51,127,110,142,69)(52,128,111,143,70)(53,113,112,144,71)(54,114,97,129,72)(55,115,98,130,73)(56,116,99,131,74)(57,117,100,132,75)(58,118,101,133,76)(59,119,102,134,77)(60,120,103,135,78)(61,121,104,136,79)(62,122,105,137,80)(63,123,106,138,65)(64,124,107,139,66), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,63)(34,64)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(97,160)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,105)(2,112)(3,103)(4,110)(5,101)(6,108)(7,99)(8,106)(9,97)(10,104)(11,111)(12,102)(13,109)(14,100)(15,107)(16,98)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,135)(34,142)(35,133)(36,140)(37,131)(38,138)(39,129)(40,136)(41,143)(42,134)(43,141)(44,132)(45,139)(46,130)(47,137)(48,144)(65,92)(66,83)(67,90)(68,81)(69,88)(70,95)(71,86)(72,93)(73,84)(74,91)(75,82)(76,89)(77,96)(78,87)(79,94)(80,85)(113,152)(114,159)(115,150)(116,157)(117,148)(118,155)(119,146)(120,153)(121,160)(122,151)(123,158)(124,149)(125,156)(126,147)(127,154)(128,145), (1,47,85,18,151)(2,48,86,19,152)(3,33,87,20,153)(4,34,88,21,154)(5,35,89,22,155)(6,36,90,23,156)(7,37,91,24,157)(8,38,92,25,158)(9,39,93,26,159)(10,40,94,27,160)(11,41,95,28,145)(12,42,96,29,146)(13,43,81,30,147)(14,44,82,31,148)(15,45,83,32,149)(16,46,84,17,150)(49,125,108,140,67)(50,126,109,141,68)(51,127,110,142,69)(52,128,111,143,70)(53,113,112,144,71)(54,114,97,129,72)(55,115,98,130,73)(56,116,99,131,74)(57,117,100,132,75)(58,118,101,133,76)(59,119,102,134,77)(60,120,103,135,78)(61,121,104,136,79)(62,122,105,137,80)(63,123,106,138,65)(64,124,107,139,66), (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,129)(28,130)(29,131)(30,132)(31,133)(32,134)(33,63)(34,64)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(43,57)(44,58)(45,59)(46,60)(47,61)(48,62)(65,87)(66,88)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,95)(74,96)(75,81)(76,82)(77,83)(78,84)(79,85)(80,86)(97,160)(98,145)(99,146)(100,147)(101,148)(102,149)(103,150)(104,151)(105,152)(106,153)(107,154)(108,155)(109,156)(110,157)(111,158)(112,159) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,105),(2,112),(3,103),(4,110),(5,101),(6,108),(7,99),(8,106),(9,97),(10,104),(11,111),(12,102),(13,109),(14,100),(15,107),(16,98),(17,55),(18,62),(19,53),(20,60),(21,51),(22,58),(23,49),(24,56),(25,63),(26,54),(27,61),(28,52),(29,59),(30,50),(31,57),(32,64),(33,135),(34,142),(35,133),(36,140),(37,131),(38,138),(39,129),(40,136),(41,143),(42,134),(43,141),(44,132),(45,139),(46,130),(47,137),(48,144),(65,92),(66,83),(67,90),(68,81),(69,88),(70,95),(71,86),(72,93),(73,84),(74,91),(75,82),(76,89),(77,96),(78,87),(79,94),(80,85),(113,152),(114,159),(115,150),(116,157),(117,148),(118,155),(119,146),(120,153),(121,160),(122,151),(123,158),(124,149),(125,156),(126,147),(127,154),(128,145)], [(1,47,85,18,151),(2,48,86,19,152),(3,33,87,20,153),(4,34,88,21,154),(5,35,89,22,155),(6,36,90,23,156),(7,37,91,24,157),(8,38,92,25,158),(9,39,93,26,159),(10,40,94,27,160),(11,41,95,28,145),(12,42,96,29,146),(13,43,81,30,147),(14,44,82,31,148),(15,45,83,32,149),(16,46,84,17,150),(49,125,108,140,67),(50,126,109,141,68),(51,127,110,142,69),(52,128,111,143,70),(53,113,112,144,71),(54,114,97,129,72),(55,115,98,130,73),(56,116,99,131,74),(57,117,100,132,75),(58,118,101,133,76),(59,119,102,134,77),(60,120,103,135,78),(61,121,104,136,79),(62,122,105,137,80),(63,123,106,138,65),(64,124,107,139,66)], [(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,129),(28,130),(29,131),(30,132),(31,133),(32,134),(33,63),(34,64),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(43,57),(44,58),(45,59),(46,60),(47,61),(48,62),(65,87),(66,88),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(73,95),(74,96),(75,81),(76,82),(77,83),(78,84),(79,85),(80,86),(97,160),(98,145),(99,146),(100,147),(101,148),(102,149),(103,150),(104,151),(105,152),(106,153),(107,154),(108,155),(109,156),(110,157),(111,158),(112,159)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 10 | 40 | 2 | 5 | 5 | 8 | 40 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D10 | C4○D16 | D4×D5 | D5×D8 | SD32⋊3D5 |
| kernel | SD32⋊3D5 | D5×C16 | C16⋊D5 | C5⋊D16 | C5⋊Q32 | C5×SD32 | D8⋊3D5 | Q8.D10 | C5⋊2C8 | C4×D5 | SD32 | Dic5 | D10 | C16 | D8 | Q16 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 |
Matrix representation of SD32⋊3D5 ►in GL4(𝔽241) generated by
| 138 | 41 | 0 | 0 |
| 200 | 138 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 214 | 156 | 0 | 0 |
| 156 | 27 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 177 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 0 | 240 | 190 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [138,200,0,0,41,138,0,0,0,0,1,0,0,0,0,1],[214,156,0,0,156,27,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,51,240,0,0,1,0],[0,64,0,0,177,0,0,0,0,0,240,0,0,0,190,1] >;
SD32⋊3D5 in GAP, Magma, Sage, TeX
{\rm SD}_{32}\rtimes_3D_5 % in TeX
G:=Group("SD32:3D5"); // GroupNames label
G:=SmallGroup(320,543);
// by ID
G=gap.SmallGroup(320,543);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,135,184,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^7,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations